(In reply to

More results and solution by goFish)

I cant get the straights to count themselves, maybe you can point out where I've muddled up.

I approached this problem by considering three separate situations, one
where you were dealt 2 wild cards, one where you were dealt 1 wild
card, and one where you were dealt no wild cards. I then
pro-rated the calculated probabilities based on the respective
probability of having been dealt that many wild cards.

The 2 blocks of text below are the brute force combinations that give a
straight with one and two wild cards, suits not withstanding. 'X'
is used to indicate a wild card, and the sets are ordered in decreasing
order of natural cards.

A,K,Q,J,X

A,K,Q,X,T

A,K,X,J,T

A,X,Q,J,T

X,K,Q,J,T

K,Q,J,X,9

K,Q,X,T,9

K,X,J,T,9

X,Q,J,T,9

Q,J,T,X,8

Q,J,X,9,8

Q,X,T,9,8

X,J,T,9,8

J,T,9,X,7

J,T,X,8,7

J,X,9,8,7

X,T,9,8,7

T,9,8,X,6

T,9,X,7,6

T,X,8,7,6

X,9,8,7,6

9,8,7,X,5

9,8,X,6,5

9,X,7,6,5

X,8,7,6,5

8,7,6,X,4

8,7,X,5,4

8,X,6,5,4

X,7,6,5,4

7,6,5,X,3

7,6,X,4,3

7,X,5,4,3

X,6,5,4,3

6,5,4,X,2

6,5,X,3,2

6,X,4,3,2

X,5,4,3,2

5,4,3,X,A

5,4,X,2,A

5,X,3,2,A

X,4,3,2,A

A,K,Q,X,X

A,K,X,J,X

A,K,X,X,T

A,X,Q,J,X

A,X,Q,X,T

A,X,X,J,T

X,K,Q,J,X

X,K,Q,X,T

X,K,X,J,T

K,Q,X,X,9

K,X,J,X,9

K,X,X,T,9

X,X,Q,J,T

X,Q,J,X,9

X,Q,X,T,9

Q,J,X,X,8

Q,X,T,X,8

Q,X,X,9,8

X,X,J,T,9

X,J,T,X,8

X,J,X,9,8

J,T,X,X,7

J,X,9,X,7

J,X,X,8,7

X,X,T,9,8

X,T,9,X,7

X,T,X,8,7

T,9,X,X,6

T,X,8,X,6

T,X,X,7,6

X,X,9,8,7

X,9,8,X,6

X,9,X,7,6

9,8,X,X,5

9,X,7,X,5

9,X,X,6,5

X,X,8,7,6

X,8,7,X,5

X,8,X,6,5

8,7,X,X,4

8,X,6,X,4

8,X,X,5,4

X,X,7,6,5

X,7,6,X,4

X,7,X,5,4

7,6,X,X,3

7,X,5,X,3

7,X,X,4,3

X,X,6,5,4

X,6,5,X,3

X,6,X,4,3

6,5,X,X,2

6,X,4,X,2

6,X,X,3,2

X,X,5,4,3

X,5,4,X,2

X,5,X,3,2

5,4,X,X,A

5,X,3,X,A

5,X,X,2,A

X,X,4,3,2

X,4,3,X,A

X,4,X,2,A

X,X,3,2,A

These two sets count properly (42 with 1X and 63 with 2X) when
determining how many straight flushes there are, but do not do so with
regular straights, according to the pointed website statistics.

For clarity, there are 40 straight flushes possible with no wild cards,
332 (42*4suits*2wilds) with one wild card and 252 (63*4suits) with 2
wild cards, totalling 624, as expected.

With no wild cards, there are 10240 (10 possible sets *4suits *4suits *
4suits * 4suits * 4suits) possible straights, subtracting the 40
straight flushes gives 10200 ranked as a straight (which agrees with
the conventional numbers). This logic should carry forward to the
one and two wild card cases, but I can't make it do so.

My math gives, for 1 wild card, 42*4*4*4*4*2 total straights (21504)
with 1 wild and 63*4*4*4 total striaghts (4032) with 2 wilds.
Subtracting the 332+252 straight flushes and adding the 10200
from the no wild calculations gives 35152, not 34704 as indicated by
the website and agreed by goFish, a difference of 448.

I've looked and looked and looked, but can't determine why I don't agree...